T-Table (Critical Values)

How to read the t-table

1. Find your degrees of freedom. For a one-sample or paired t-test, df = n − 1. For a pooled two-sample test, df = n₁ + n₂ − 2. Welch's test uses a fractional df; round it down.
2. Choose the column from your significance level and tails. Columns are labelled by the area in one upper tail, with the matching two-sided confidence level underneath. A two-sided test at α = 0.05 uses the 0.025 column (95% CI). A one-sided test at α = 0.05 uses the 0.05 column.
3. Compare: if the absolute value of your t statistic is larger than the table value, the result is significant at that level.

Worked example: a paired t-test

Twelve students take a test before and after a revision course, and a paired t-test on their score differences gives t = 2.45. With n = 12 pairs, df = 11. For a two-sided test at α = 0.05, the 0.025 column in row 11 gives 2.201. Because 2.45 > 2.201, the improvement is statistically significant at the 5% level. (The exact two-sided p-value is 0.032; the t-test calculator computes it directly.)

Using the table for a confidence interval

A sample of 16 measurements has mean 52 and standard deviation 6. The standard error is 6 / √16 = 1.5, and the 95% critical value for df = 15 is 2.131. The margin of error is 2.131 × 1.5 = 3.20, so the 95% confidence interval is 48.8 to 55.2.

When your df is not in the table

The table jumps from 30 to 40, 50, 60, 80, 100 and 120. If your df falls between rows, use the next smaller df: its critical value is slightly larger, so your conclusion stays on the safe (conservative) side. For df = 45, use row 40 (2.021); the exact value is 2.014. As df grows, the t-distribution approaches the normal, which is why the last row equals the z critical values (1.960 for 95%).

Common mistakes

• Using the 0.05 column for a two-sided test at 5%. That column puts all 5% in one tail and gives a too-small critical value (1.725 instead of 2.086 at df = 20).
• Using n instead of n − 1 for the degrees of freedom of a one-sample or paired test.
• Treating paired data as two independent samples. Before-and-after measurements on the same people need the paired test with df = number of pairs − 1.

Need more precision than a table gives? The p-value calculator returns the exact p-value for any t and df, and how to calculate a p-value shows the method.

See also: Critical value calculator · How to do a t-test by hand

Frequently asked questions

How do you read a t-table?

Find the row for your degrees of freedom and the column for your significance level and number of tails. If the absolute value of your t statistic exceeds the table value, the result is significant at that level.

Which column do I use for a two-tailed test at 0.05?

The column for 0.025 in one tail, which is also labelled as the 95% confidence level. At df = 20 the critical value is 2.086.

What if my degrees of freedom are not in the table?

Use the next smaller degrees of freedom. Its critical value is slightly larger, which keeps the conclusion conservative. For example, use df = 40 for df = 45.

Why does the last row of the t-table match the z-table?

As degrees of freedom increase, the t-distribution approaches the standard normal distribution, so with infinite degrees of freedom the critical values equal the z values such as 1.960.